We show that Kirchhoff ’s law of conservation holds for non-commutative graph flows if and only if the graph is planar. We generalize the theory of (Euclidean) lattices to infinite dimension and... Show moreWe show that Kirchhoff ’s law of conservation holds for non-commutative graph flows if and only if the graph is planar. We generalize the theory of (Euclidean) lattices to infinite dimension and consider the ring of algebraic integers as such a lattice. We compute some invariants using capacity theory and obtain a partial solution to the (algorithmic) closest vector problem. We generalize the results on (universally) graded rings by Lenstra and Silverberg. We study the special case of group rings, and show that under similar assumptions rings can be uniquely decomposed into a group ring in a maximal way. We give a functorial algorithm to compute roots of fractional ideals of orders in number rings. Show less
In this work we consider a generalization of graph flows. A graph flow is, in its simplest formulation, a labelling of the directed edges with real numbers subject to various constraints. A common... Show moreIn this work we consider a generalization of graph flows. A graph flow is, in its simplest formulation, a labelling of the directed edges with real numbers subject to various constraints. A common constraint is conservation in a vertex, meaning that the sum of the labels on the incoming edges of this vertex equals the sum of those on the outgoing edges. One easy fact is that if a flow is conserving in all but one vertex, then it is also conserving in the remaining one. In our generalization we do not label the edges with real numbers, but with elements from an arbitrary group, where this fact becomes false in general. As we will show, graphs with the property that conservation of a flow in all but one vertex implies conservation in all vertices are precisely the planar graphs. Show less
Bergh, M.J.H. van den; Castelein, S.T.; Gent, D.M.H. van 2020
The positional game of Order versus Chaos can be considered a maker-breaker variant. The players Order and Chaos take turns placing circles or crosses on a board, in which the goal of Order is to... Show moreThe positional game of Order versus Chaos can be considered a maker-breaker variant. The players Order and Chaos take turns placing circles or crosses on a board, in which the goal of Order is to create a consecutive line of identical symbols of a certain length, while Chaos aims to prevent this. In this paper, we provide some theoretical results on winning strategies for both players on finite boards of varying sizes, as well as on infinite boards. The composition of these strategies was aided by the use of Monte-Carlo Tree Search (MCTS) players, as well as a SAT solver. In addition to these theoretical results, we provide some more experimental results obtained using MCTS. Show less
